I have very little background with functional derivatives and I would like to clarify some issues.

I am trying to compute the second functional derivative of the Klein Gordon action expressed in real components


I know how to compute the first functional derivative


which gives


from now on I have doubts. I wanna compute


If there were no derivatives this would be trivial but I don't know how to deal with $\partial^2$. I have thought in introducing a delta to write this as an integral and use the formula I have used to compute the first functional derivative, but I am concerned about taking derivatives of something having a Dirac delta.

So, my question is, how am I supposed to compute this functional derivative?

  • $\begingroup$ Everywhere replace $\phi_1(x) \to \phi_1(x) + \alpha \delta(x-y)$ in your penultimate formula, next takes the derivative $\frac{d}{d\alpha}|_{\alpha=0}$... $\endgroup$ – Valter Moretti Jun 30 '15 at 11:26
  • $\begingroup$ @ValterMoretti and why? $\endgroup$ – Yossarian Jun 30 '15 at 11:28
  • 2
    $\begingroup$ You can consider that as the definition of functional derivative... $\endgroup$ – Valter Moretti Jun 30 '15 at 11:30
  • $\begingroup$ Second functional derivatives are also discussed in e.g. this Phys.SE post. $\endgroup$ – Qmechanic Jun 30 '15 at 11:36
  • $\begingroup$ @ValterMoretti and how do I dela with the derivative of the delta? $\endgroup$ – Yossarian Jun 30 '15 at 13:31

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